fixed point theorems in random fuzzy metric space through

Computer Engineering and Intelligent Systems www.iiste.org

ISSN 2222-1719 (Paper) ISSN 2222-2863 (Online)

Vol.4, No.9, 2013

14

Fixed Point Theorems in Random Fuzzy Metric Space Through

Rational Expression R.P. Dubey

* Ramakant Bhardwaj2, Neeta Tiwari

*1, Ankur Tiwari

*C.V. Raman University, Bilaspur (C.G.)

2Dept. of Mathematics Truba institute of Engineering & IT Bhopal (MP) India

Abstract

In the present paper we will find some fixed point theorems in random fuzzy metric space, random fuzzy 2-

metric space and random fuzzy 3-metric space through rational expression.Also we will find the results for

integral type mappings.

2.1 Introduction In 1965, the concept of fuzzy set was introduced by Zadeh [39]. After him many authors have developed the

theory of fuzzy sets and applications. Especially, Deng [9], Erceg [11], Kaleva and Seikkala [26]. Kramosil and

Michalek [28] have introduced the concept of fuzzy metric spaces by generalizing the definition of probabilistic

metric space. Many authors have also studied the fixed point theory in these fuzzy metric spaces are [1], [7],

[13], [19], [21], [24], [25], [32] and for fuzzy mappings [2], [3], [4], [5], [22], [31].

In 1994, George and Veeramani [18] modified the definition of fuzzy metric spaces given by Kramosil and

Michalek [28] in order to obtain Hausdroff topology in such spaces. Gregori and Sapena [20] in 2002 extended

Banach fixed point theorem to fuzzy contraction mappings on complete fuzzy metric space in the sense of

George and Veeramani [18]. It is remarkable that Sharma, Sharma and Iseki [34] studied for the first time

contraction type mappings in 2-metric space. Wenzhi [38] and many others initiated the study of Probabilistic 2-

metric spaces. As we know that 2-metric space is a real valued function of a point triples on a set X, whose

abstract properties were suggested by the area of function in Euclidean spaces.

Now it is natural to expect 3-metric space which is suggested by the volume function. The method of introducing

this is naturally different from 2-metric space theory from algebraic topology.

The concept of Fuzzy-random-variable was introduced as an analogous notion to random variable in order to

extend statistical analysis to situations when the outcomes of some random experiment are fuzzy sets. But in

contrary to the classical statistical methods no unique definition has been established before the work of Volker

[37]. He presented set theoretical concept of fuzzy-random-variables using the method of general topology and

drawing on results from topological measure theory and the theory of analytic spaces. No results in fixed point

are introduced in random fuzzy spaces. In [17] paper authors Gupta, Dhagat, Shrivastava introduced the fuzzy

random spaces and proved common fixed point theorem.

In the present paper we will find some fixed point theorems in random fuzzy metric space, random fuzzy 2-

metric space and random fuzzy 3-metric space through rational expression.Also we will find the results for

integral type mappings.

To start the main result we need some basic definitions.

2.2 Preliminaries:

2.2.1 Definitions

Definition2.2.1.1: (Kramosil and Michalek 1975)

A binary operation *: [0,1]´[0,1] ® [0,1] is a t-norm if it satisfies the following conditions :

(i) *(1,a) = a , *(0,0) = 0

(i) *(a,b) = *(b,a)

(ii) *(c,d) ³ *(a,b) whenever c ³ a and d ³ b

(iii) *(*(a,b) ,c) = *(a, *(b,c) ) where a,b, c,dÎ [0,1]

Definition 2.2.1.2: (Kramosil and Michalek 1975)

The 3-tuple (X,M, *) is said to be a fuzzy metric space if X is an arbitrary set * is a continuous t-norm and M is

a fuzzy set on X2 ´ [0,¥) satisfying the following conditions:

(i) M(x,y,0) = 0



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(ii) M(x,y,t) = 1 for all t > 0 iff x = y,

(iii) M(x,y,t) = M(y,x,t) ,

(iv) M(x,y,t) * M(y,z,s) £ M(x,z,t + s) ,

(v) M(x,y, .) : [0,¥[®[0,1] is left-continuous,

Where x, y, z Î X and t, s > 0.

In order to introduce a Hausdroff topology on the fuzzy metric space, in (Kramosil and Michalek 1975) the

following definition was introduced.

Definition 2.2.1.3: (George and Veermani 1994)

The 3-tuple (X, M, *) is said to be a fuzzy metric space if X is an arbitrary set, * is a continuous t-norm and M is

a fuzzy set on X2 ´]0,¥[ satisfying the following conditions :

(i) M(x,y,t) > 0

(ii) M(x,y,t) = 1 iff x = y,

(iii) M(x,y,t) = M(y,x,t) ,

(iv) M(x,y,t) * M(y,z,s) £ M(x,z,t + s) ,

(v) M(x,y, .) : ]0,¥[®[0,1] is continuous,

Where x, y, z Î X and t, s > 0.

Definition 2.2.1.4: (George and Veermani 1994)

In a metric space (X, d) the 3-tuple (X, Md,*) where Md(x, y, t) = t / (t + d(x, y)) and a*b = ab is a fuzzy metric

space. This Md is called the standard fuzzy metric space induced by d.

Definition 2.2.1.5: (Gregori and Sepene 2002)

Let (X, M,*) be a fuzzy metric space. A mapping f: X→ X is fuzzy contractive if there exists 0 < k < 1 such

that

≤ k

For each x,y ϵ X and t > 0 .

Definition 2.2.1.6: (Gregori and Sepene 2002)

Let (X, M, *) be a fuzzy metric space . We will say that the sequence xn in X is fuzzy contractive if there

exists k ϵ (0, 1) such that

≤ k for all t > 0 , n ϵ N.

We recall that a sequence xn in a metric space (X, d) is said to be contractive if there exist 0 < k < 1 such that d

(xn+1, xn+2) ≤ kd (xn, xn+1) for all n ϵ N.

Definition 2.2.1.7: (Kumar and Chugh 2001)

Let (X, τ) be a topological space. Let f and g be mappings from a topological space (X, τ) into itself. The

mappings f and g are said to be compatible if the following conditions are satisfied:

(i) fx = gx , x Î X Implies fgx = gfx ,

(ii) The continuity of f at a point x in X implies lim gfxn=fx whenever xn is a sequence in X such

that lim gxn=lim fxn=fx for some x in X.

Definition 2.2.1.8 : A binary operation * : [0,1] x [0,1] x [0,1]®[0,1] is called a continuous t-norm if ([0,1], *) is

an abelian topological monoid with unit 1 such that a1* b1* c1≤ a2* b2* c2 whenever a1≤ a2, b1≤ b2, c1≤ c2 for all

a1, a2, b1, b2 and c1, c2 are in [0,1].

Definition 2.2.1.9 : The 3-tuple (X,M,*) is called a fuzzy 2-metric space if X is an arbitrary set, * is a continuous

t-norm and M is a fuzzy set in X3 x [0,¥) satisfying the following conditions for all x, y, z, u Î X and t1, t2, t3 >

0.

(FM’-1) M(x, y, z, 0) = 0,

(FM’-2) M(x, y, z, t) = 1, t > 0 and when at least two of the three points are equal,

(FM’-3) M(x, y, z, t) = M(x, z, y, t) = M(y, z, x, t),

(Symmetry about three variables)

(FM’-4) M(x, y, z, t1+t2+t3) ≥ M(x, y, u, t1)*M(x, u, z, t2)* M(u, y, z, t3)

(This corresponds to tetrahedron inequality in 2-metric space)



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The function value M(x, y, z, t) may be interpreted as the probability that the area of triangle is less than t.

(FM’-5) M(x, y, z, .): [0, 1)®[0,1] is left continuous.

Definition 2.2.1.10: Let (X, M,*) is a fuzzy 2-metric space:

(1) A sequence xn in fuzzy 2-metric space X is said to be convergent to a point x Î X, if

, x, a, t) = 1

for all aÎ X and t > 0.

(2) A sequence xn in fuzzy 2-metric space X is called a Cauchy sequence, if

, , a, t) = 1

for all aÎ X and t > 0, p > 0.

(3) A fuzzy 2-metric space in which every Cauchy sequence is convergent is said to be complete.

Definition 2.2.1.11: A function M is continuous in fuzzy 2-metric space iff whenever xn®x, yn® y, then

, , a, t) = M(x, y, a,t)

for all aÎ X and t > 0.

Definition 2.2.1.12: Two mappings A and S on fuzzy 2-metric space X are weakly commuting iff

M (ASu, SAu, a, t) ≥ M (Au, Su, a, t)

for all u, a Î X and t > 0.

Definition 2.2.1.13: A binary operation * : [0,1]4 ®[0,1] is called a continuous t-norm if ([0,1], *) is an abelian

topological monoid with unit 1 such that a1* b1* c1* d1≤ a2* b2* c2 * d2 whenever a1≤ a2, b1≤ b2, c1≤ c2 and d1≤

d2for all a1, a2, b1, b2 , c1, c2 and d1, d2 are in [0,1].

Definition 2.2.1.14 : The 3-tuple (X, M, *) is called a fuzzy 3-metric space if X is an arbitrary set, * is a

continuous t-norm and M is a fuzzy set in X4 x [0,¥) satisfying the following conditions : for all x, y, z, w, u ÎX

and t1, t2, t3, t4 > 0.

(FM’’-1) M(x, y, z, w, 0) = 0,

(FM’’-2) M(x, y, z, w, t) = 1 for all t > 0,

(only when the three simplex á x, y, z, w ñ degenerate)

(FM’’-3) M(x, y, z, w, t) =M(x, w, z, y, t) = M(y, z, w, x, t) = M (z, w, x, y, t) = …

(FM’’-4) M(x, y, z, w, t1 +t2 + t3 + t4) ≥ M(x, y, z, u, t1)*M(x, y, u, w, t2)

*M(x, u, z, w, t3)*M(u, y, z, w, t4)

(FM’’-5) M(x, y, z, w, .) : [0,1)®[0,1] is left continuous.

Definition 2.2.1.15: Let (X, M, *) be a fuzzy 3-metric space:

(1) A sequence xn in fuzzy 3-metric space x is said to be convergent to a point x Î X, if

, x, a, b, t) = 1

for all a, b Î X and t > 0.

(2) A sequence xn in fuzzy 3-metric space X is called a Cauchy sequence, if

, , a, b, t) = 1

for all a, b Î X and t > 0, p > 0.

(3) A fuzzy 3-metric space in which every Cauchy sequence is convergent is said to be complete.

Definition 2.2.1.16: A function M is continuous in fuzzy 3-metric space iff whenever xn®x, yn®y

, , a, b, t) = M(x, y, a, b, t)

for all a, b Î X and t > 0.

Definition 2.2.1.17: Two mappings A and S on fuzzy 3-metric space X are weakly commuting iff

M (ASu, SAu, a, b, t) ≥ M (Au, Su, a, b, t)

for all u, a, b Î X and t > 0.



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Definition2.2.1.18: Throughout this chapter, denotes a measurable space is a measurable

selector. X is any non empty set. is continuous t-norm, M is a fuzzy set in

A binary operation *:[0,1]x[0,1]®[0,1] is called a continuous t-norm if ([0,1],*) is an abelian

Topological monodies with unit 1 such that a * b ≥ c * d whenever

a ≥ c and b ≥ d , For all a, b, c, d, Î [0, 1]

Example of t-norm are a * b = a b and a * b = min a, b

Definition2.2.1.18 (a): The 3-tuple (X, M, W *) is called a Random fuzzy metric space, if X is an arbitrary

set,* is a continuous t-norm and M is a fuzzy set in X2 x [0,¥) satisfying the following conditions: for all

x x, x y, x z Î X and s, t > 0,

( )

( )

( )

( )

[ ) [ ]

( 1) : , ,0 0

( 2) : , , 1, 0,

( 3) : , , ( , , )

( 4) : , , ( , , ) ( , , )

( 5) : ( , , ) : 0,1 0,1

RFM M x y

RFM M x y t t x y

RFM M x y t M y x t

RFM M x z t s M x y t M z y s

RFM M x y a is left continuous

x x

x x

x x x x

x x x x x x

x x x

- =

- = " Û =

- =

- + ³ *

- ®

0 x0, xx0,

In what follows, (X, M, W ,*) will denote a random fuzzy metric space. Note that M (x x, x y, t) can be thought

of as the degree of nearness between x x and x y with respect to t. We identify x x = x y with M (x x, x y, t)

= 1 for all t > 0 and M (x x, x y, t) = 0 with ¥.In the following example, we know that every metric induces a

fuzzy metric.

Example Let (X, d) be a metric space.

Define a *b = a b, or ab =min a, b) and for all x, y, Î X and t > 0,

( )( )

, ,,

tM x y t

t d x yx x

x x=

+

Then (X, M, W , *) is a fuzzy metric space. We call this random fuzzy metric M induced by the metric d the

standard fuzzy metric.

Definition2.2.1.18 (b): Let (X, M, W , *) is a random fuzzy metric space.

(i)A sequence x xn in X is said to be convergent to a point x x Î X,

lim ( , , ) 1nn

M x x tx x®¥

=

(ii) A sequence x xn in X is called a Cauchy sequence if

lim ( , , ) 1, 0 0n p nn

M x x t t and px x+®¥

= " 0and p00

(iii) A random fuzzy metric space in which every Cauchy sequence is convergent is said to be

Complete.

Let (X.M,*) is a fuzzy metric space with the following condition.

(RFM-6) lim ( , , ) 1, ,t

M x y t x y Xx x x x e®¥

= "



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. Definition2.2.1.18 (c): A function M is continuous in fuzzy metric space iff whenever

, lim ( , , ) ( , , )n n n nn

x x y y M x y t M x y tx x x x x x x x®¥

® ® Þ ®

Definition2.2.1.18 (d): Two mappings A and S on fuzzy metric space X are weakly commuting iff

M ( ASx u, SAx u, t) ≥ M (Ax u, Sx u, t)

Some Basic Results 2.2.1.18 (e):

Lemma (i) [Motivated by 19] for all x x, x y, Î X, M(ξx, ξy) is non -decreasing.

Lemma (ii) Let ξyn be a sequence in a random fuzzy metric space (X, M, Ω,*) with the condition

(RFM -6) If there exists a number q Î (0,1) such that

2 1 1( , , ) ( , , ) , 0 1,2,3......,

.

n n n n

n

M y y qt M y y t t and n

then y is a cauchy sequencein X

x x x x

x

+ + +³ " =and n0000

Lemma (iii) [Motivated by 32] If, for all ξx, ξy Î X, t > 0 and for a number q Î (0,1),

( , , ) ( , , ),M x y qt M x y t then x yx x x x x x³ =

Lemmas 1, 2, 3 of 2.2.1.18 (e): ) hold for random fuzzy 2-metric spaces and random fuzzy 3-metric spaces

also.

Definition2.2.1.18 (f): A binary operation *: [0, 1] x [0,1] x [0,1] ® [0,1] is called a continuous t-norm if

([0,1],*) is an abelian topological monodies with unit 1 such that a1 * b1 * c1 ≥ a2 * b2 * c2 whenever

a1 ≥ a2, b1 ≥ b2, c1≥ c2 for all a1, a2, b1, b2 and c1, c2 are in [0,1].

Definition2.2.1.18 (g): The 3-tuple (X, M, *) is called a random fuzzy 2-metric space if X is an arbitrary set,

* is continuous t-norm and M is fuzzy set in X3

x [0,¥) satisfying the followings

( )

( )

( )

( )

'

'

'

'

1, 2, 3, 1, 2,

( 1) : , , ,0 0

( 2) : , , , 1, 0,

( 3) : , , ( , , , ) ( , , , ), var

( 4) : , , , , ( , , , ) ( , , , ) ( , , ,

RFM M x y z

RFM M x y z t t x y

RFM M x y t M x z y t M y z x t symmetry about three riable

RFM M x y z t t t M x y u t M x u z t M u y z

x x x

x x x

x x x x x x x x

x x x x x x x x x x x x

- =

- = " Û =

- = =

- ³ * *

0 x0, xx0,

[ ) [ ]

3,

'

1 2 3

)

( 5) : ( , , ) : 0,1 0,1 , , , , , , , 0

t

RFM M x y z is left continuous x y z u X t t tx x x x x x x e- ® " 0

.

Definition2.2.1.18 (h): Let (X, M, * ) be a random fuzzy 2-metric space. A sequence x xn in fuzzy 2-metric

space X is said to be convergent to a point x x Î X,

lim ( , , , ) 1, 0nn

M x x a t for all a X and tx x x x e®¥

= 0

(2) A sequence x xn in random fuzzy 2-metric space X is called a Cauchy sequence, if

lim ( , , , ) 1, , 0n p nn

M x x a t for all a X and t px x x x e+®¥

= 0

(3) A random fuzzy 2-metric space in which every Cauchy sequence is convergent is said to be complete.

Definition2.2.1.18 (i): A function M is continuous in random fuzzy 2-metric space, iff whenever



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For all x a Î X and t > 0.

, , lim ( , , , ) ( , , , ), 0n n n nn

x x y y then M x y a t M x y a t a X and tx x x x x x x x x x x e®¥

® ® = " 0

Definition2.2.1.18 (j): Two mappings A and S on random fuzzy 2-metri space X are weakly commuting iff

( , , , ) ( , , , ), , 0M AS u SA u a t M A u S u a t u a Xand tx x x x x x x x e³ " 0

Definition2.2.1.18 (k): A binary operation *: [0, 1]4 ® [0, 1] is called a continuous t-norm if

([0, 1], *) is an abelian topological monoid with unit 1 such that

22221111 dcbadcba ***³*** Whenever a1 ≥ a2, b1 ≥ b2 , c1 ≥ c2 and d1 ≥ d2 for all a1, a2, b1, b2, c1, c2

and d1, d2 are in [0,1].

Definition2.2.1.18 (l): The 3-tuple (X, M, Ω, *) is called a fuzzy 3-metric space if X is an arbitrary set, * is a

continuous t-norm monoid and M is a fuzzy set in X4 x [0, ¥] satisfying the following conductions:

( )

( )

( )

( )

''

''

''

''

2 3

( 1) : , , , ,0 0

( 2) : , , , , 1, 0,

, , , deg

( 3) : , , , , ( , , , , ) ( , , , , )

( 4) : , , , , ( , ,

RFM M x y z w

RFM M x y z w t t

Only whenthethreesimplex x y z w enerate

RFM M x y z w t M x w z y t M z w x y t

RFM M x y z w t t t M x y

x x x x

x x x x


x x x x x x

- =

- = "

- = = = - - - -

- + + ³

0,

[ ) [ ]

1,

2, 3, 4

''

1 2 3 4

, , )

( , , , , ) ( , , , , ) ( , , , , )

( 5) : ( , , , ) : 0,1 0,1 ,

, , , , , , , , 0

z u t

M x y u w t M x u z w t M u y z w t

RFM M x y z w is left continuous

x y z u w X t t t t

x x


x x x x

x x x x x e

*

* *

- ®

" 0

Definition2.2.1.18 (m): Let (X, M, Ω,*) be a Random fuzzy 3-metric space:

(1)A sequence ξXn in fuzzy 3-metric space X is said to be convergent to a point ξx Î X, if

lim ( , , , , ) 1, , 0nn

M x x a b t for all a b X and tx x x x x x e®¥

= 0

(2)A sequence x xn in random fuzzy 3-metric space X is called a Cauchy sequence, if

lim ( , , , , ) 1, , , 0n p nn

M x x a b t for all a b X and t px x x x x x e+®¥

= 0

(3)A random fuzzy 3-metric space in which every Cauchy sequence is convergent is said to be complete.

Definition2.2.1.18 (n): A function M is continuous in random fuzzy 3-metric space if

, , lim ( , , , , ) ( , , , ), , 0n n n nn

x x y y then M x y a b t M x y a t a b X and tx x x x x x x x x x x x x e®¥

® ® = " 0

Definition2.2.1.18 (o): Two mappings A and S on random fuzzy 3-metric space X are weakly commuting iff,

( , , , , ) ( , , , , ) , , 0M AS u SA u a b t M A u S u a b t u a b Xand tx x x x x x x x e³ " 0

2.2.2 Prepositions.

Preposition 2.2.2.1 (Gregori and Sepene 2002)

Let (X, d) be a metric space. The mapping f: X à X is a contractive (a contraction) on the metric space (X, d)

with contractive constant k iff f is fuzzy contractive, with contractive constant k, on the standard fuzzy metric

space (X, Md, * ), induced by d.



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20


Let (X, M, *) be a complete fuzzy metric space in which fuzzy contractive sequences are Cauchy. Let T: X à

X be a fuzzy contractive mapping being k the contractive constant. Then T has a unique fixed point.


Let (X, Md, * ) be the standard fuzzy metric space induced by the metric d on X. The sequence xn in X is

contractive in (X, d) iff xn is fuzzy contractive in (X, Md,*).

Preposition 2.2.2.1 and 2.2.2.3 imply that Preposition 2.2.2.2 is a generalization of Banach fixed point theorem

to fuzzy metric spaces as defined by George and Veermani.

It is to be noted that all the prepositions are true for (RFM)

Now, we state and prove our main theorem as follows,

2.3 Main Results

Theorem 2.3.1: Let be a complete Random fuzzy metric space in which fuzzy contractive sequences

are Cauchy and T , R and S be mappings from into itself is a measurable selector

satisfying the following conditions :

T(X) Í R(X) and T(X) Í S(X) (2.3.1.1)

- 1≤ k k (2.3.1.2)

with 0 < k < 1 and

Q( x, y, t) = min

The pairs T, S and T, R are compatible. R, T and S are w-continuous. (2.3.1.3)

Then R, T and S have a unique common fixed point. (2.3.1.4)

Proof: Let x0 Î X be an arbitrary point .Since T(X) Í R(X) and T(X) Í S(X), we can construct a

sequence xn in X such that

T = RRRRRR = S (2.3.1.5)

Now,

Q ( ) = min

= min

= min

We now claim that

Otherwise we claim that

i.e Q ( ) = (2.3.1.6)

\ ≤ k [by 2.3.1.2]

which is a contradiction.

Hence, ≤ k (2.3.1.7)

\ T xn is a fuzzy contractive sequence in So T xn is a Cauchy sequence

As X is a complete fuzzy metric space, T xn-1 is convergent. So, T xn-1 converges to some point z in X.

\ T xn-1, R xn, S xn converges to z. By w-continuity of R, S and T, there exists a point u in X such that

xn ® u as n®¥ and so = = = z implies

RRRRRRRRRRRRRRR u = S u = T u = z (2.3.1.8)

Also by compatibility of pairs T, S and T, R and Tu = Ru = Su = z implies

T z = TRTRTRTRTRTR u = RT u = RRRRRR z and T z = TS u = ST u = S z



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Therefore, T z = RRRRRRRRRRRRRR z = S z (2.3.1.9)

We now claim that T z = z.

If not ≤ k

Q ( z, u, t) = min

= min

= min

=

\ ≤ k


Hence T z = z

So z is a common fixed point of R, T and S.

Now suppose v ≠ z be another fixed point of R, T and

\ ≤ k

Q ( v, u, t) = min

= min

= min

=

\ ≤ k

which is a contradiction. Hence v = z.

Thus R, T and S have a unique fixed point.

Theorem 2.3.2: Let be a complete Random fuzzy 2-metric space (RF-2M) in which fuzzy

contractive sequences are Cauchy and T , R and S be mappings from into itself is a

measurable selector and a( )=a > 0satisfying the following conditions :

T(X) Í R(X) and T(X) Í S(X) (2.3.2.1)

- 1≤ k k k k k (2.3.2.2)

with 0 < k < 1 and

Q( x, y, a, t) = min



Proof: Let x0 Î X be an arbitrary point of X. Since T(X) Í R(X) and T(X) Í S(X), we can construct a


T = RRRRRR = S (2.3.2.5)

Now,

Q( =



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=min

= min

We now claim that


i.e. Q ( ) = (2.3.2.6)

\ ≤ k [by (2.3.2.2)]


Hence, ≤ k (2.3.2.7)

\ T xn is a fuzzy contractive sequence in (X, M,*). So T xn is a Cauchy sequence in

As X is a complete Random fuzzy2- metric space, T xn-1 is convergent. So, T xn-1 converges to some point

z in X.

\ T xn-1, R xn, S xn converges to z. By w-continuity of R, S and T, there exists a point u in X such

that xn ® u as n®¥ and so = = = z implies

RR u = S u = T u = z (2.3.2.8)

Also by compatibility of pairs T, S and T, R and T u = RR u = S u = z implies

T z = TRTR u = RT u = RR z and T z = TS u = ST u = S z

Therefore, T z = RR z = S z (2.3.2.9)


If not ≤ k

Q ( z, u, a, t) = min

= min

= min

=

\ ≤ k


Hence T z = z



\ ≤ k

Q( v, u, a, t) = min

= min

= min

=



Vol.4, No.9, 2013

23

\ ≤ k

This is a contradiction. Hence v = z.

Thus R, T and S have unique common fixed point. This completes our proof.



measurable selector and , > 0satisfying the following conditions :

T(X) Í R(X) and T(X) Í S(X) (2.33.1)

- 1≤ k k k k k (2.3.3.2)

with 0 < k < 1 and

Q( x, y, a, t) = min





T = RRRRR = S (2.3.3.5)

Now,

Q( =

=min

= min

We now claim that


i.e. Q ( ) = (2.3.3.6)

\ ≤ k [by (2.3.32)]


Hence, ≤ k (2.3.3.7)



z in X.



RR u = S u = T u = z (2.3.3.8)



Therefore, T z = RR z = S z (2.3.39)


If not ≤ k

Q ( z, u, a, t) =



Vol.4, No.9, 2013

24

min

= min

= min

=

\ ≤ k


Hence T z = z



\ ≤ k

Q( v, u, a, t) = min

= min

= min

=

\ ≤ k



2.4 Integral type mapping

Impact of fixed point theory in different branches of mathematics and its applications is immense. In 2002, A.

Branciari [3] analyzed the existence of fixed point for mapping f defined on a complete metric space (X, d)

satisfying a general contractive condition of integral type.

THEOREM 2.4.1: (Branciari)

Let (X, d) be a complete metric space, c Î(0, 1) and let f: X®X be a mapping such that for each x, y Î X,

≤ c (2.4.1.1)

where x : [0,+¥)®[0,+¥) is a lebesgue integrable mapping which is summable on each compact subset of

[0,+¥), non-negative and such that for each e > 0, , then f has a unique fixed point a Î X such that for

each x Î X, = a.

After the paper of Branciari, a lot of research works have been carried out on generalizing

contractive condition of integral type for a different contractive mapping satisfying various known properties.

Theorem 2.4.2: Let (X, d) be a complete metric space and f : X®X such that

+

For each x, yÎX with non-negative real’s a, b, g such that 2a + b + g < 1, where u :[0, +¥)®[0, +¥)

is a Lebesgue integrable mapping which is summable , non-negative and such that for each e > 0,

> 0

Then f has a unique fixed point in X.

There is a gap in the proof of theorem (3.4.2.1) . In fact, the authors [10] used the inequality ≤

+ for 0 ≤ a < b, which is not true in general .The aim of research paper of H. Aydi [23] was

to present in the presence of this inequality an extension of theorem 2.4.2 using altering distances .



Vol.4, No.9, 2013

25

THEOREM 2.4.3: Let (X, M, *) be a complete Random Fuzzy metric space in which fuzzy contractive

sequences are Cauchy and T, R and S be mappings from (X, M, *) into itselflflflflflflflf denotes a measurable

space is a measurable selector satisfying the following conditions:

T( X ) Í R( X ) and T( X ) Í S( X )

- 1≤ k k k

with 0 < k < 1 and

=

For every Where is a legesgue

integrable mapping which is summable on each compact subset of , non negative, and such

that,

The pairs T, S and T, R are compatible, R, T, S are w-continuous. Then T, R and S have a unique common fixed

point.

Proof: Let x0 Î X be an arbitrary point .Since T(X) Í R(X) and T(X) Í S(X), we can construct a sequence

xn in X such that

T = RRRRR = S

Now

=

=

We now claim that


i.e. ,

\ Hence, ≤ k

\ \ T xn is a fuzzy contractive sequence in So T xn is a Cauchy sequence


\ T xn-1, R xn, S xn converges to z. By w-continuity of R, S and T, there exists a point u in X such that

xn ® u as n®¥ and so = = = z implies



Vol.4, No.9, 2013

26

RRRRRRRRRRRR u = S u = T u = z

Also by compatibility of pairs T, S and T, R and Tu = Ru = Su = z implies

T z = TRTRTRTRTRTR u = RT u = RRRRRR z and T z = TS u = ST u = S z

Therefore, T z = RRRRRRRRRRRRRRR z = S z


If not ≤ k

=

=

=

=

\ ≤ k


Hence T z = z



\ ≤ k

=

=

=

\ ≤ k


Thus R, T and S have a unique fixed point.



measurable selector and a( ) >0 satisfying the following conditions :



Vol.4, No.9, 2013

27

T(X) Í R(X) and T(X) Í S(X) (2.4.2.1)

- 1≤ k k k k (2.4.2.2)

with 0 < k < 1 and

=



that,


point.

PROOF: Let x0Î X be an arbitrary point of X. Since T(X)Í R(X) and T(X)Í S(X) , we can construct a


Txn-1 = Rxn = Sxn

Now ,

=

=

We now claim that


i.e. , ≤ k [by (2.4.2.2)]

which is a contradiction

\ -1 ≤ k

This is a contradiction.

Hence, Hence, ≤ k

T xn is a fuzzy contractive sequence in (X, M,* ). So Txn is a Cauchy sequence in

(X, M,* ).


\ T xn-1, RR xn, S xn converges to z. By w-continuity of R, S and T, there exists a point u in X such


RR u = S u = T u = z (2.4.2.8)



Vol.4, No.9, 2013

28



Therefore, T z = RR z = S z (2.4.2.9)


If not ≤ k

=

=

=

=

=

≤ k


Hence T z = z


Now suppose v ≠ z be another fixed point of R, T and S


\ ≤ k

=

=

=

\ \ ≤ k




measurable selector and , > 0satisfying the following conditions :

T(X) Í R(X) and T(X) Í S(X) (2.43.1)



Vol.4, No.9, 2013

29

- 1≤ k k k k k (2.4.3.2)

with 0 < k < 1 and

=



that,


point.



T = RRRRRR = S (2.4.3.5)

Now,

=

N We now claim that


i.e. ,

\ ≤ k [by (2.4.3.2)]

This is a contradiction.

Hence, ≤ k (2.4.3.7)



z in X.



RR u = S u = T u = z (2.4.3.8)



Therefore, T z = RR z = S z




Vol.4, No.9, 2013

30

If not ≤ k

=

=

=

=

\ ≤ k


Hence T z = z



\ ≤ k

=

=

=

=

≤ k



Acknowledgement: Authors are thankful to MPCST for the project No. 2556.”Application of Fixed point theory

in Pure and Applied Mathematics”. This is research cum review article authors are thankful to those authors

whose references are used.

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